An Introduction to Functional Programming Through Lambda Calculus

More documents

Recommendations

Info

- 78 -def 3 = SUCC 2etc.For example, 1 expands as:First of all:SUCC 0 => ... =>if isnumb 0then MAKE_NUMB (succ (value N))else NUMB_ERRORisnumb 0 ==equal (type 0) numb_type ==equal (λobj.(obj select_first) 0) numb_type =>equal (0 select_first) numb_type ==equal (λs.(s numb_type zero) select_first) numb_type -> ... ->equal numb_type numb_type => ... =>trueThus, we next evaluate:MAKE_NUMB (succ (value 0)) ==MAKE_NUMB (succ (λobj.(obj select_second) 0)) ->MAKE_NUMB (succ (0 select_second)) ==MAKE_NUMB (succ (λs.(s numb_type zero) select_second) -> ... ->MAKE_NUMB (succ zero) ==MAKE_NUMB one ==λvalue.λs.(s numb_type value) one =>λs.(s numb_type one)In general, a typed number is a pair with the untyped equivalent as value.We can now redefine the predecessor function to return an error for zero:def PRED N =if isnumb Nthenif iszero (value N)then NUMB_ERRORelse MAKE_NUMB ((value N) select_second)else NUMB_ERROR
- 79 -Note that we return NUMB_ERROR for a non-number argument and a zero number argument. We could construct moreelaborate error objects to distinguish such cases but we won’t pursue this further here.We will need a typed test for zero:def ISZERO N =if isnumb Nthen MAKE_BOOL (iszero (value N))else NUMB_ERRORNow we can redefine the binary arithmetic operations. They all need to test that both arguments are numbers so wewill introduce an auxiliary function to do this:def both_numbs X Y = and (isnumb X) (isnumb Y)Now for addition based on our earlier definition:def + X Y =if both_numbs X Ythen MAKE_NUMB (add (value X) (value Y))else NUMB_ERRORand multiplication:def * X Y =if both_numbs X Ythen MAKE_NUMB (mult (value X) (value Y))else NUMB_ERRORand division to take account of a zero divisor:def / X Y =if both_numbs X Ythenif iszero (value Y)then NUMB_ERRORelse MAKE_NUMB (div1 (value X) (value Y))else NUMB_ERRORand equality:def EQUAL X Y =if both_numbs X Ythen MAKE_BOOL (equal (value X) (value Y))else NUMB_ERROR5.9. CharactersLet us now add characters to our types. Character values are specified exhaustively:’0’ is a character’1’ is a character...’9’ is a character’A’ is a character’B’ is a character
Page 1 and 2:
AN INTRODUCTION TO FUNCTIONAL PROGR
Page 3 and 4:
- 3 -LISP was one of the earliest l
Page 5 and 6:
- 5 -1.4. Execution order in impera
Page 7 and 8:
- 7 -B[1] + B[2] + ... + B[M] + SUM
Page 9 and 10:
- 9 -quantifiers to describe proper
Page 11 and 12:
- 11 -program specification all use
Page 13 and 14:
- 13 -To begin with, we will briefl
Page 15 and 16:
- 15 -Now, compare this with the fo
Page 17 and 18:
- 17 -wherefor example: ::= λx.x
Page 19 and 20:
- 19 -giving:λx.xAn identity opera
Page 21 and 22:
- 21 -λx.xThis is now evaluated as
Page 23 and 24:
- 23 -in the body:λarg.(func arg)g
Page 25 and 26:
- 25 -We can use the self applicati
Page 27 and 28: - 27 -and body:λsecond.firstWhen a
Page 29 and 30: - 29 -2.12.3. Making pairs from two
Page 31 and 32: - 31 -(λx.x λx.x) =>λx.xIt is po
Page 33 and 34: - 33 -i) the expression is an appli
Page 35 and 36: - 35 -((λfunc.λarg.(func arg) arg
Page 37 and 38: - 37 -Bound variablesi) is bound i
Page 39 and 40: - 39 -i) λx.λy.(λx.y λy.x)ii)
Page 41 and 42: - 41 -(((cond false) true) x) ==(((
Page 43 and 44: - 43 -3.5. OROR is a binary operato
Page 45 and 46: - 45 -two ==(succ one) ==(λn.λs.(
Page 47 and 48: - 47 -So:def pred = λn.(((cond zer
Page 49 and 50: - 49 -For example, we could re-writ
Page 51 and 52: - 51 -4) Show that:λx.(succ (pred
Page 53 and 54: - 53 -gobble a sweet andgobble a sw
Page 55 and 56: - 55 -then xelse f (succ x) (pred y
Page 57 and 58: - 57 -and:-> ... ->will still imply
Page 59 and 60: - 59 -If this function is now appli
Page 61 and 62: - 61 -def add1 f x y =if iszero yth
Page 63 and 64: - 63 -the second minus the first wi
Page 65 and 66: - 65 -succ (succ (div1 one four)) -
Page 67 and 68: - 67 -n * n-1 * n-2 * ... * 1in nor
Page 69 and 70: - 69 -within our intended interpret
Page 71 and 72: - 71 -Sometimes it may be necessary
Page 73 and 74: - 73 -λvalue.λs.(s error_type val
Page 75 and 76: - 75 -def AND X Y =if and (isbool X
Page 77: - 77 -def ISBOOL B = MAKE_BOOL (isb
Page 81 and 82: - 81 -def ’1’ = MAKE_CHAR (succ
Page 83 and 84: - 83 -will do so. However, for a no
Page 85 and 86: - 85 -To simplify the presentation
Page 87 and 88: - 87 -rec 0 = or (SUCC ) = Thus,
Page 89 and 90: - 89 -def signed_type = ...def SIGN
Page 91 and 92: - 91 -If the head of a list and the
Page 93 and 94: - 93 -def MAKE_LIST = make_obj list
Page 95 and 96: - 95 -LIST_ERRORThe test for an emp
Page 97 and 98: - 97 -CONS 1 (APPEND (TAIL (CONS 1
Page 99 and 100: - 99 -For example:1::(2::(3::(4::[]
Page 101 and 102: - 101 -LIST_EQUAL [] [] = TRUELIST_
Page 103 and 104: - 103 -In general, the string:" "is
Page 105 and 106: - 105 -(STRING_LESS (’r’::"ridg
Page 107 and 108: - 107 -gives value of "" with 10*(1
Page 109 and 110: - 109 -In general, for:rec =IF IS
Page 111 and 112: - 111 -Note that this description t
Page 113 and 114: - 113 -Similarly, to remove a speci
Page 115 and 116: - 115 -rec DOUBLE [] = []or DOUBLE
Page 117 and 118: - 117 -"cats"::"dogs"::(λW.(APPEND
Page 119 and 120: - 119 -‘TAIL ’>6.18. Exercises1
Page 121 and 122: - 121 -["Betty","Baker"].For exampl
Page 123 and 124: - 123 -For example:IF STRING_EQUAL
Page 125 and 126: - 125 -["VDU",25,12]::["modem",20,1
Page 127 and 128: - 127 -matches the argument:so:[,,]
Page 129 and 130:
- 129 -For example, suppose we have
Page 131 and 132:
- 131 -to place the Nth item, skip
Page 133 and 134:
- 133 -[7,EMPTY,EMPTY]To add 4 to t
Page 135 and 136:
- 135 -[3,EMPTY,EMPTY],[5,EMPTY,[6,
Page 137 and 138:
- 137 -TFIND 5 [4,[3,EMPTY,EMPTY],[
Page 139 and 140:
- 139 -7.14. Binary tree efficiency
Page 141 and 142:
- 141 -λ[a,b].(SUM_SQ2 a b)so:def
Page 143 and 144:
- 143 -and so on.def istype (t::o)=
Page 145 and 146:
- 145 - ::= ( + ) |( + ) |( + ) |(
Page 147 and 148:
- 147 -IF ISZERO (ADD 1 2)THEN 1ELS
Page 149 and 150:
- 149 -cond one (add (succ one) (pr
Page 151 and 152:
- 151 -λdummy.e2Instead, we have t
Page 153 and 154:
- 153 -The first Church-Rosser theo
Page 155 and 156:
- 155 -(λx.x λy.y) 2=>λy.yand th
Page 157 and 158:
- 157 -For example, consider:SQUARE
Page 159 and 160:
- 159 -SQUARE 1 => ... =>IFIND 1 SQ
Page 161 and 162:
- 161 - : For objects which do not
Page 163 and 164:
- 163 -is a tuple consisting of two
Page 165 and 166:
- 165 -to indicate both integer and
Page 167 and 168:
- 167 -- tl;> fn : (’a list) -> (
Page 169 and 170:
- 169 -Functions have the form:fn
Page 171 and 172:
- 171 -For example, to find the lar
Page 173 and 174:
- 173 -9.19. Pattern matchingSML fu
Page 175 and 176:
- 175 -end;> val sort = fn : (int l
Page 177 and 178:
- 177 -- fun length [] = 0 |length
Page 179 and 180:
- 179 -- datatype traffic_light = r
Page 181 and 182:
- 181 -but this defines the new typ
Page 183 and 184:
- 183 -con empty = empty : (’a tr
Page 185 and 186:
- 185 -iv)Join strings s1 and s2 to
Page 187 and 188:
- 187 -( + )( - )( * )( / ) == numb
Page 189 and 190:
- 189 -10.4. Logictis the primitive
Page 191 and 192:
- 191 -(> 9 8 7 6 5) ->tThe primiti
Page 193 and 194:
- 193 -It is important to note that
Page 195 and 196:
- 195 -nilwhich may be also written
Page 197 and 198:
- 197 -(car ’(1 2 3)) ->1(cdr ’
Page 199 and 200:
- 199 -(tadd 7 nil) ->(7 nil nil)(t
Page 201 and 202:
- 201 -For example, suppose the ave
Page 203 and 204:
- 203 -10.17. Symbols, quoting and
Page 205 and 206:
- 205 -Note once again that there m
Page 207 and 208:
- 207 -iv) find the sum of applying
Page 209 and 210:
- 209 -ii) Write a function which i
Page 211 and 212:
- 211 -Chapter 3 - Conditions, bool
Page 213 and 214:
- 213 -11. R. M. Burstall, J. S. Co
Page 215 and 216:
- 215 -55. P. Wegner, Programming L
Page 217 and 218:
- 217 - - (g h) - - g - - h - f
Page 219 and 220:
- 219 -λb.b3)i) a) (identity ) =>
Page 221 and 222:
- 221 -λb.(λa.{a} b))λa.{a}a fre
Page 223 and 224:
- 223 -not (or false false) => ...
Page 225 and 226:
- 225 -false (implies false true) f
Page 227 and 228:
- 227 -then fun nelse add (fun n) (
Page 229 and 230:
- 229 -i) ISBOOL 3 => ... =>MAKE_BO
Page 231 and 232:
- 231 -iii)def SIGN_+ X Y =if and (
Page 233 and 234:
- 233 -THEN [H,M,S]ELSElet M = M +
Page 235 and 236:
- 235 -λy.y5 reductionsλf.λa.(f
Page 237 and 238:
- 237 -gconstr (v:int) ((h::t):int
Page 239 and 240:
- 239 -0(+ n (nsum (- n 1)))))ii) (
Page 241:
- 241 -(if (= m1 m2)(if (< s1 s2)tn
show all

An Introduction to Functional Programming Through Lambda Calculus

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?